11Txil = T Lixll, T > 0; (4) Llxi + X211 < Llxlx1 + L1x211
VOL. 27, 1941 MA THEMA TICS: D. G. BOURGIN 539 be a fixed chain in A"(K, G). In the direct sumZ,, + 1(K, I) + G, the ele- ments of the form 5d, - z-d., for all chains d., constitute a subgroup S iso- morphic with i + 1(K, 1). The difference group (E. +1 + G) - S = E(xf) is a group extension of G by the cohomology group I + 1(K, I) = a+1 - i+. 1 Zassenhaus, H., Lehrbuch der Gruppentheorie, Berlin, 1937, pp. 89-98. 2 Pontrjagin, L., Topological Groups, Princeton, 1939, Ch. V; v. Kampen, E. R., Ann. Math., 36, 448-463 (1935). ' Whitney, H., Duke Math. Jour., 4, 495-528 (1938). 4 Steenrod, N. E., Amer. Jour. Math., 58, 661-701 (1936). 5 Tucker, A. W., Ann. Math., 34, 191-243 (1933). 6 This topology will be a Hausdorf topology if G is a division closure group (Steenrod, loc. cit., p. 676). 7 tech, E., Fund. Math., 25, 33-44 (1935). 8 Lefschetz, S., Topology, New York, 1930, p. 325. * Steenrod, N. E., Ann. Math., 41, 833-851 (1940). 10 This is essentially the Steenrod universal coefficient theorem for compact metric spaces. SOME PROPERTIES OF REAL LINEAR TOPOLOGICAL SPACES BY D. G. BOURGIN DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ILLINOIS Communicated September 26, 1941 We denote a real linear topological space, 1. t. s., by the symbol L and adopt Hyers" slight modification of the von Neumann axioms2 for the characterization of such a space. If 9 = {N) is a neighborhood basis of the origin,0I, in the1.
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